Here by a representation of \((X,A)\)  as a \(G\) -fiberwise NDR pair we understand, in analogy with [1]}, that there is a pair
\((u,h)\)  of maps with \(u\colon X\rightarrow I\)  and \(h\colon X\times _B I\rightarrow X\)  which represent \((X,A)\)  as a fiberwise
NDR pair and which satisfy \(u(xg) = u(x)\)  for all \(x\in X\)  and \(g\in G\) , and \(h(xg,t) = h(x,t)g\)  for all
\((x,t)\in X\times _B I\)  and \(g\in G\) .

We will explain how to adapt Steps 1–3 in the proof of Theorem 4.4.4
of [2]} to our setting. Step 3 adapts in a straightforward
way to show that \(i(A)\)  is closed in \(X\) :
factor the inclusion \(i\colon A\rightarrow X\)  as \(A\rightarrow E\rightarrow X\)  where
\(E = A\times _B I \cup X\times _B (0,1]\)  and where
\(i_0\colon A\rightarrow E\)  is given by \(i_0(a) = (a,0)\) . Analogous to
the corresponding statement in
[2]}, the projection \(\pi \colon E\rightarrow X\)  is an \(f\) -acyclic
\(f\) -fibration in \(G{K}/B\) . Therefore there exists a
map \(\lambda \colon X\rightarrow E\)  extending \(i\) , i.e. \(\lambda \circ i = i_0\) .
Let \(\psi \colon E\rightarrow I\)  denote the projection onto the second factor and
note that \(\psi ^{-1}(0) = i_0(A)\) , so that \(i_0(A)\)  is closed in \(E\) .
Therefore \(\lambda ^{-1}i_0(A) = i(A)\)  is closed in \(X\)  (since \(\lambda \)  is injective). Standard
arguments now show that \((X,A)\)  has a representation as a \(G\) -fiberwise NDR pair.
